Download Effects of ionospheric damping on MHD wave mode structure D.-H. Lee

E-LETTER
Earth Planets Space, 56, e33–e36, 2004
Effects of ionospheric damping on MHD wave mode structure
D.-H. Lee1 , K.-H. Kim1 , R. E. Denton2 , and K. Takahashi3
1 Department
of Astronomy and Space Science, Kyung Hee University, Korea
of Physics and Astronomy, Dartmouth College, USA
3 The Johns Hopkins University Applied Physics Laboratory, USA
2 Department
(Received June 19, 2004; Revised October 20, 2004; Accepted October 21, 2004)
We calculate the mode structure of magnetospheric MHD waves on a meridional plane. We have added the
effect of ionospheric dissipation to the three-dimensional dipole field MHD model of Lee and Lysak (1999); this
model allows a realistic Alfven speed profile for the plasmasphere and realistic boundary conditions at the outer
boundaries that vary with respect to local time. Using power spectra and plots of spatial mode structure, we show
that the two-dimensional transverse modes on the dipolar meridian are strongly affected by ionospheric damping,
but the compressional modes are not. The location of field line resonances spreads wide as the damping increases,
but the compressional mode structure remains stable.
Key words: MHD waves, magnetosphere, ionosphere, field line resonance.
1.
Introduction
Magnetohydrodynamic waves with wavelengths more
than a few Earth radii can provide useful information about
global features of the magnetosphere since the wave phenomena depend strongly on the inhomogeneity and geometry of the whole magnetosphere. Shear Alfven waves, which
are called field line resonances (FLR), enable us to study the
density distribution along the magnetic field lines by monitoring standing mode frequencies (Baransky et al., 1985;
Waters et al., 1995; Chi and Russell, 1998; Denton et al.,
2001). Compressional waves, which are strongly affected
by the surrounding magnetospheric boundaries, can excite
Pi 2 pulsations and/or Pc 5 waves in the magnetotail (Yeoman and Orr, 1989; Sutcliffe and Yumoto, 1991; Takahashi
et al., 1995; Lee and Kim, 1999).
The magnetosphere is a highly nonuniform medium for
MHD waves, and the amplitude, frequency, and polarization
of ULF waves observed on the ground and in space strongly
depend on the location of measurements. The ionospheric
effects also play an significant role in determining the mode
structure and observational feature of ULF waves. It has
been studied how FLR structure and phases along the magnetic field lines are affected by the finite ionospheric conductivities (e.g., Hughes and Southwood, 1976; Newton et al.,
1978; Allan and Knox, 1979; Allan, 1982; Ellis and Southwood, 1983; Budnik et al., 1998; Yoshikawa et al., 1999;
Southwood and Kivelson, 2001). Therefore, it is important
to investigate the details of mode structure using a realistic
model. In this study, we present the two-dimensional mode
structure using a three-dimensional dipole MHD model (Lee
and Lysak, 1999). Effects of different values of ionospheric
damping are examined by means of the power spectra and
c The Society of Geomagnetism and Earth, Planetary and Space Sciences
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(SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan;
The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB.
spatial mode structure of each MHD mode. In particular, the
mode structure in this numerical experiment will be useful
in understanding satellite observations of FLR and Pi2 pulsations (e.g., Osaki et al., 1998; Takahashi et al., 2003).
2.
Model
In Lee and Lysak’s dipole field MHD model (1989), the
linearized MHD wave equations in a cold plasma are solved
by a leapfrog scheme. It is assumed that the inner boundary is at L = 2 and the outer boundary is at L = 10.5,
and perfect reflecting boundary conditions are used at the
ionospheres, which are assumed to be at an altitude of 0.5
RE . (For more details about the model, see Lee and Lysak
(1999).)
Figure 1 shows the Alfven speed profile at the equator
used in the model, which is based on observations (e.g.,
Chappell, 1988). The density along each field line is assumed to vary as r −6 , following Lee and Lysak (1989),
where r is the geocentric distance. This assumption enables us to obtain relatively constant Alfven speeds along
each magnetic field line, which is useful for numerical convenience to avoid large computing time at the high-latitude
polar region.
Ionospheric damping is a new feature of the model used in
this paper. The damping coefficient d = 1 − E ref /E in , where
E in and E ref represent the incident and reflected wave amplitudes, respectively. Thus d = 0 corresponds to a perfect
conducting boundary condition (no dissipation), and d = 1
corresponds to the free boundary condition (100% dissipation or no reflected waves). For instance, whenever the initial waves with the amplitude E o arrive at the ionosphere, the
reflected wave amplitude would be (1 − d)E o .
Since the magnetosphere is not a perfect reflector and
also asymmetric in local time, we apply different boundary
conditions at the dayside and nightside outer regions. At the
dayside magnetopause the Alfven speed drops from Vms ≈
e33
e34
D.-H. LEE et al.: MHD WAVE MODE STRUCTURE
Fig. 1. The equatorial Alfven speed assumed in the model. The dashed and
dash-dot curves are at L values corresponding to maximum and minimum
Alfven speed, respectively.
600 km/s in the magnetosphere to Vsw ≈ 60 km/s in the solar
wind (in fact, the outer region of the magnetosphere is the
magnetosheath, where the Alfven speed is a little larger than
that of the solar wind. Since it is still very small compared
to the magnetospheric Alfven speed, we neglect such details
in this model). Thus the reflection coefficient at the dayside
boundary (Lee and Lysak, 1999) is given by
r=
Vms − Vsw
Vms + Vsw
(1)
for waves propagating out of the magnetosphere. In the
nightside region, compressional waves can propagate without significant speed changes through L = 10.5 (the outer
boundary of our model), so we use a free boundary there.
The regions between the dayside and nightside at dawn and
dusk have reflection coefficients which vary gradually from r
in (1) on the dayside to zero on the nightside (Lee and Lysak,
1999). Both at the ionosphere and the outer boundary in this
model, the reflected waves are determined by the coefficients
d and r .
3.
Numerical Results
We start the simulation with a radially inward propagating impulse in the magnetotail that is symmetric in latitude
(thus only odd harmonic waves are excited). This impulse
was used in Lee and Lysak (1999), where you find more details about the impulse. We will look at the wave spectra and
spatial mode structure for different damping coefficients: no
damping (d = 0.0), intermediate damping (d = 0.1), and
large damping (d = 0.3). Figure 2 shows the wave spectra
near the equator and the meridional spatial mode structure
of the electric field components for d = 0. The left and the
right panels represent the spectra at noon and midnight, respectively. To avoid the antinode or node regions, spectra are
calculated at grid points lying on a line that is slightly off the
equatorial plane. Thus, please note that the spectral feature
in Figs. 2–4 is not necessarily corresponding to the mode
structure across the magnetic shells at the exact equator (see
figure 6 of Budnik et al. (1998)).
The radial electric field component effectively shows the
shear Alfven waves or transverse waves, while the east-west
component shows the compressional waves. As confirmed
in Lee and Lysak (1999) (see their figure 3 for the magnetic
field spectra when d = 0.0 is assumed), the nightside equa-
Fig. 2. No ionospheric damping is assumed (d = 0.0): (a) the power spectra
of the electric field components at the equator and (b) the meridional
spatial mode structure for f = 8 mHz. The dashed and dash-dot curves
are plotted at L values corresponding to maximum and minimum Alfven
speed in Fig. 1, respectively.
torial region has nearly harmonic peaks in the east-west component (compressional mode) around f = 8, 16–23, 30, 40
mHz, and so on. These peaks show up most distinctly inside
the plasmapause; the outer magnetosphere has less distinctive spectral features. For both power spectra corresponding
to the dayside in Fig. 2(a), some of plasmaspheric modes are
still found and relatively weak harmonic structure appears in
the outer region. This occurs owing to the Alfven speed gradients at the dayside magnetopause. In Fig. 2(b), the meridional spatial mode structure for f = 8 mHz is presented as
an example. The mode structure is obtained by calculating
the Fourier amplitude of f = 8 mHz at each point in space
during the period of 1000 s after the impulse. Relatively
strong transverse waves that are coupled to the f = 8 mHz
compressional mode appear both in the inner and outer regions (radial component). Note that the initial impulse was
assumed to be symmetric with respect to the equator, so only
the fundamental mode and its odd harmonics are excited.
The nightside east-west electric field (compressional
mode) in Fig. 2(b) has relatively large amplitude just inside
the plasmapause where the Alfven speed becomes lowest
(Lee and Kim, 1999). This is understandable considering
that the electric field for this lowest frequency harmonic is
likely to have largest power in the region corresponding to
the minimum Alfven speed that is surrounded by “walls”
with larger Alfven speed (Lee and Kim, 1999). It is interesting to note that this mode is not confined within the
plasmasphere and the amplitude is rather extended beyond
the plasmapause (or beyond the region of maximum Alfven
speed). This fact is consistent with the properties of plasma-
D.-H. LEE et al.: MHD WAVE MODE STRUCTURE
Fig. 3. Same as Fig. 2 except that intermediate ionospheric damping is
assumed (d = 0.1).
Fig. 4. Same as Fig. 2 except that strong ionospheric damping is assumed
(d = 0.3).
4.
spheric virtual resonances (Lee and Kim, 1999; Fujita et al.,
2001) that arise owing to imperfect reflection at the plasmapause.
Figure 3 shows the case of d = 0.1. In Fig. 3(a), the
continuous frequency bands of the radial component (transverse mode) become relatively wide compared to Fig. 2(a).
For instance, it is found that the spectrum becomes relatively
broadband in the ambient region near L = 6. However, there
are no significant changes in the spectral features of the eastwest component (compressional mode). This point is consistent with the mode structure displayed in Fig. 3(b). The
compressional mode structure remains almost the same as
that shown in Fig. 2(b) in both global shape and amplitude.
However, the transverse mode structure that was previously
localized becomes spread over neighboring magnetic shells,
and the amplitude significantly decreases compared to the
case of no ionospheric damping.
Figure 4 shows the case of d = 0.3. The continuous
frequency bands of the radial component (transverse mode)
power spectrum become even thicker, while the spectrum
of the east-west component (compressional mode) seems
similar to those of Figs. 2 and 3. This feature is also observed in the spatial mode structure. The east-west component (compressional mode) remains close to that in Figs. 2
and 3, including the amplitude. Therefore, it is evident
from Figs. 2(b)–4(b) that the amplitude of the radial component (transverse waves) significantly decreases and the radial
width of the spectral features increases as the damping coefficient increases, while the east-west component (compressional mode) amplitude is fairly stable for different damping
coefficients.
e35
Discussion and Summary
The meridional spatial mode structure presented above is
limited to the case of f = 8 mHz. However, the features
distinguishing transverse and compressional mode structure
for f = 8 mHz are also found at other spectral peaks, such
as f = 16–23, 30, 40 mHz (not shown). In both power
spectrum and spatial structure, transverse modes are strongly
affected by the damping rate, while the compressional modes
are not. The results shown here are also indicative of the
effects of other ionospheric damping coefficients, such as
d = 0.8, 0.5, and so on.
In eigenmode analysis of coupled wave equations, damping is often expressed using a complex frequency ω =
ωr + iγ . For finite γ , it is expected that both transverse
and compressional waves will decay via dissipation. As γ
increases, the spectral peaks become more broadband while
the peak frequency ωr remains the same. This feature is consistent with our numerical results above in Figs. 3 and 4. It
should be noted that γ represents an average value resulting
from a Fourier transform over infinite time. In a very long
simulation, both transverse and compressional modes must
eventually have the same damping rate. However, our studies suggest that the damping rates are very different when
the damping is assumed only at the ionospheres in a finitetime simulation. This differential time-dependent behavior
will be important in interpreting MHD wave properties associated with ionospheric damping.
We assumed that the incident waves at the ionosphere
are reflected with the reduced amplitude for a given damping rate. In general, the reflection coefficient can be complex if we include the phase relation at the boundary (e.g.,
Yoshikawa et al., 1999). We neglected such phase variations
at the ionosphere for simplicity in this study. In addition,
e36
D.-H. LEE et al.: MHD WAVE MODE STRUCTURE
ionospheric damping rates are assumed to be uniform in our Hughes, J. W. and D. J. Southwood, The screening of micropulsation signals
by the atmosphere and ionosphere, J. Geophys. Res., 81, 3234, 1976.
model. In reality, the nightside damping should be more sigLee, D. H. and K. Kim, Compressional MHD waves in the magnetosphere:
nificant than the dayside damping. Effects of such nonuniA new approach, J. Geophys. Res., 104, 12,379, 1999.
form ionospheric conductivities remain as future work.
Lee, D. H. and R. L. Lysak, Magnetospheric ULF wave coupling in the
Acknowledgments. Work at Kyung Hee University was supported
by the Korea Science & Engineering Foundation grant R14-2002043-01000-0. Work at Dartmouth College was supported by NSF
grant ATM-0245664.
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